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arXiv · 2204.02476

Local lens rigidity for manifolds of Anosov type

Mihajlo Cekić·Colin Guillarmou·Thibault Lefeuvre
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Abstract

The lens data of a Riemannian manifold with boundary is the collection of lengths of geodesics with endpoints on the boundary together with their incoming and outgoing vectors. We show that negatively-curved Riemannian manifolds with strictly convex boundary are locally lens rigid in the following sense: if $g_0$ is such a metric, then any metric $g$ sufficiently close to $g_0$ and with same lens data is isometric to $g_0$, up to a boundary-preserving diffeomorphism. More generally, we consider the same problem for a wider class of metrics with strictly convex boundary, called metrics of Anosov type. We prove that the same rigidity result holds within that class in dimension $2$ and in any dimension, further assuming that the curvature is non-positive.

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BibTeXRIS

Mihajlo Cekić, Colin Guillarmou, Thibault Lefeuvre. 2022-04-05. Local lens rigidity for manifolds of Anosov type. https://arxiv.org/abs/2204.02476

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